Transverse modes

Transverse modes are the patterns of the electric field across a beam, perpendicular to its direction of travel, that a laser cavity or a waveguide supports and that keep their shape as they propagate. In laser beams they are labeled TEM${}_{mn}$: the fundamental TEM₀₀ mode has a Gaussian profile with no nodes and the best possible beam quality, $M^2=1$, while a higher-order mode has $m$ and $n$ dark nodal lines across the beam, a larger size and a faster divergence, with $M^2=2m+1$ and $2n+1$ in the two axes. Each transverse mode of a cavity has its own comb of longitudinal modes, at frequencies set by the cavity length and the curvature of its mirrors. In optical fibers and waveguides the corresponding modes are labeled differently, LP${}_{lm}$ in fiber, with LP${}_{01}$ the single mode of a single-mode fiber.

Hermite-Gaussian modes

A stable two-mirror cavity, or any system with rectangular symmetry, supports the Hermite-Gaussian modes, whose field is

$$\begin{array}{rl}{\displaystyle E_{mn}(x,y)\propto }& {\displaystyle H_m\text{\hspace{0.17em}⁣}\left(\frac{\sqrt{2}x}{w}\right)H_n\text{\hspace{0.17em}⁣}\left(\frac{\sqrt{2}y}{w}\right)}\\ {\displaystyle }& {\displaystyle \times \exp \text{\hspace{0.17em}⁣}(-\frac{x^2+y^2}{w^2}),}\end{array}$$

where $H_m$ is the Hermite polynomial of order $m$ and $w$ is the $1\mathrm{/}{e}^2$ radius of the fundamental Gaussian beam at the same position. Every mode of the set shares that $w$, the same wavefront curvature and the same waist position, so they all propagate together through the cavity; the indices count the zeros of the field along $x$ and $y$.

The second-moment radius of a mode grows as $w\sqrt{2m+1}$ along $x$, and its far-field divergence grows by the same factor, so their product, the beam parameter product, is $2m+1$ times that of TEM₀₀. That factor is the mode's $M^2$, and the Gaussian beam calculator propagates a beam of any $M^2$ through focus.

Laguerre-Gaussian modes

In a cavity with circular symmetry, the natural modes are the Laguerre-Gaussian modes,

$$E_{pl}(r,\varphi )\propto {\left(\frac{\sqrt{2}r}{w}\right)}^{\mathrm{\mid }l\mathrm{\mid }}{L}_p^{\mathrm{\mid }l\mathrm{\mid }}\text{\hspace{0.17em}⁣}\left(\frac{2r^2}{w^2}\right)\exp \text{\hspace{0.17em}⁣}(-\frac{r^2}{w^2})e^{il\varphi },$$

with radial index $p=0,1,2,\dots$, counting dark rings, and azimuthal index $l=0,\pm 1,\pm 2,\dots$ Modes with $l\mathrm{\ne }0$ have a helical wavefront, a dark center and an orbital angular momentum of $l\mathrm{\hslash }$ per photon, and their beam quality is $M^2=2p+\mathrm{\mid }l\mathrm{\mid }+1$. The LG${}_{0,\pm 1}$ mode is the doughnut beam that lasers with circular symmetry sometimes produce, historically written TEM${}_{01}^{\ast }$; it is a superposition of TEM₁₀ and TEM₀₁ with a quarter-period phase difference. Doughnut beams for STED microscopy and for optical tweezers are usually made outside the laser, by passing a Gaussian beam through a spiral phase plate or a spatial light modulator.

Mode frequencies

A higher-order mode accumulates more Gouy phase between the mirrors than TEM₀₀, so it resonates at a slightly different frequency. In a two-mirror cavity of length $L$ with mirror radii $R_1$ and $R_2$,

$$\begin{array}{c}{\displaystyle {\nu }_{qmn}=\frac{c}{2L}[q+(m+n+1)\frac{\zeta }{\pi }],}\\ {\displaystyle \zeta =\arccos (\pm \sqrt{g_1{g}_2}),g_i=1-\frac{L}{R_i},}\end{array}$$

where $\zeta$ is the Gouy phase accumulated in one pass from mirror to mirror, $q$ is the longitudinal mode number and the sign follows that of $g_1$ and $g_2$; for Laguerre-Gaussian modes $m+n$ is replaced by $2p+\mathrm{\mid }l\mathrm{\mid }$. The offset between transverse orders is therefore a fixed fraction of the longitudinal mode spacing:

In the confocal case, every transverse mode falls on the frequency of a longitudinal mode or halfway between two, which is why confocal scanning Fabry-Perot interferometers work without careful mode matching, with an effective free spectral range of $c\mathrm{/}4L$; the Fabry-Perot cavity calculator covers the plane-parallel and confocal cases. In a laser, a higher-order mode running alongside TEM₀₀ shows up as extra lines in a scanning interferometer and as extra beat notes on a photodiode; in the 30 cm example above, TEM₀₁ beats with TEM₀₀ at 92 MHz and at 407 MHz, next to the 500 MHz longitudinal beat.

Why lasers are built for TEM₀₀

A TEM₀₀ beam focuses to the smallest spot, spreads least for a given size, couples into single-mode fiber with almost no loss, and keeps a predictable Gaussian profile. Lasers are made to run on it by giving the higher-order modes more loss than the fundamental, which is possible because those modes are wider:

• An aperture inside the cavity, sized to pass the TEM₀₀ beam, clips the higher-order modes.
• In end-pumped solid-state lasers, a pumped region matched to the TEM₀₀ mode size leaves the wider modes without enough gain.
• The cavity design sets the TEM₀₀ size; a cavity whose TEM₀₀ mode is small compared with the aperture of the gain medium leaves gain at the edges for higher-order modes and tends to run multimode.
• Outside the laser, a spatial filter, a pinhole at a focus, or a short length of single-mode fiber removes the higher-order content from a beam.

Transverse modes in laser diodes

Edge-emitting laser diodes are single mode in the fast axis, perpendicular to the junction, because their waveguide is made thin enough, usually a fraction of a micrometer, to guide only one mode in that direction. Across the junction, a ridge or buried waveguide a few micrometers wide keeps them in one lateral mode up to a power limit; above it, a higher-order lateral mode or a sideways shift of the fundamental appears as a kink in the light-current curve. Broad-area lasers, with stripes tens to hundreds of micrometers wide, run on many lateral modes and give high power at the cost of a slow-axis beam far from diffraction-limited. VCSELs with apertures much larger than a few micrometers also run on several transverse modes.

Multimode beams

A beam carrying several transverse modes at different frequencies is an incoherent mixture, and its $M^2$ along each axis is the power-weighted average of the individual values: a laser running on TEM₀₀ and TEM₁₀ with equal power has $M_x^2=2$ and $M_y^2=1$, and one running on modes up to TEM₃₃ has at most $M^2=7$ in each axis. High-power industrial lasers often trade beam quality for power in this way, and the pump diodes of fiber lasers deliver multimode light into large pump claddings.

Modes of waveguides and fibers

An optical waveguide also has a discrete set of transverse modes, set by its core size, index contrast and wavelength rather than by mirrors. In a weakly guiding fiber they are the linearly polarized LP${}_{lm}$ modes; LP${}_{01}$ is the fundamental, close to Gaussian, and a step-index fiber guides only this mode when its V-number is below 2.405. A slab waveguide guides TE${}_m$ and TM${}_m$ modes, with $m$ the number of field zeros across the slab, which the Waveguide Mode Explorer solves exactly for any thickness and index contrast; a channel waveguide on a chip, confined in both directions, guides quasi-TE and quasi-TM modes. A multimode fiber guides hundreds of modes, which travel at different speeds and give it its modal dispersion.

Common questions

What is the difference between transverse and longitudinal modes?

Transverse modes are field patterns across the beam, set by the mirror curvatures, apertures or waveguide that confine it. Longitudinal modes are the frequencies the cavity resonates at, set by its length. Each transverse mode has its own comb of longitudinal modes, and a laser can run on one transverse mode and several longitudinal modes at once.

What does TEM₀₀ mean?

Transverse electromagnetic mode with zero nodes in each direction across the beam: the fundamental Gaussian mode, with $M^2=1$. The TEM₀₀ mode entry gives its beam parameters.

How can you tell which transverse mode a laser is running on?

Look at the beam on a camera or a beam profiler, where the nodal lines of a higher-order mode are visible, and measure $M^2$, which is 1 for a pure TEM₀₀ beam. A higher-order mode running together with TEM₀₀ also produces beat notes on a fast photodiode at frequencies offset from the longitudinal mode spacing.

Why do higher-order modes diverge faster?

Their field extends further from the axis and has more zeros, and both the beam width and the far-field angle scale as $\sqrt{2m+1}$ in each axis. The product of size and divergence is therefore $M^2=2m+1$ times that of a Gaussian beam, so a higher-order beam of the same size spreads faster.

References: A. E. Siegman, Lasers (University Science Books, 1986). H. Kogelnik and T. Li, "Laser beams and resonators," Applied Optics 5, 1550 (1966). B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019). L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Physical Review A 45, 8185 (1992).